The best introduction I know to the entire field of topology is John McCleary's A First Course in Topology: Continuity and Dimension.Not only does it present all the essentials in a strongly geometric manner in low dimensions,it gives a historical perspective on the subject.
What's the best introduction to algebraic topology?
Well,depends on if you like geometric intuition or not. If so,Allen Hatcher's textbook is considered by many to be the new gold standard. And best of all,it's available online for free at Hatcher's website.
If you like more modern (i.e. abstract) approaches,the book by Joseph Rotman can't be beat. And recently,an awesome text by Tammo tom Dieck came out which is probably the state of the art right now and is very readable.
A book that's probably too difficult to use as a textbook but is so beautiful that it needs to be used as a supplement is Peter May's A Concise Course In Algebraic Topology.By May's own admission,it's probably too tough for a first course on the subject,but it is beautifully written and gives a great overview of the subject. It also has a very good bibliography for further study.
My favorite texts on algebraic topology?Probably the 2 books by V.V. Pravalov, Elements of Combinatorial And Differential Topology and Elements of Homology Theory.both available in hardcover from the AMS.Together,they probably give the single most complete presentation of topology that currently exists,with plenty of low-dimensional pictures,concrete constructions and emphasis on manifolds.
And of course,I'd be remiss if I didn't mention the wamdering oddball text which a lot of US universities are afraid to use for thier first course,but is a treasure trove for mathematics students: John Stillwell's Classical Topology And Combinatorial Group Theory. An incredibly rich historical presentation by a master. It's strange organization and selection of material is a double edged sword-but it will give amazing insight to the basic ideas of topology and how they develped. These tools will give a student coming out of Stillwell's book a very strong foundation for studying more modern presentations. I highly recommend it to anyone interested in topology at any level.
There are also several terrific free online lecture sources you should look at,primarily the complete notes of K.Wurthmuller and Gregory Naber. Both can be found at Math Online and I recommend them both highly.